On Ratio Monotonicity of a New Kind of Numbers Conjectured by Z.-W. Sun

TL;DR

This paper proves a new monotonicity conjecture related to ratios of a sequence of positive integers, using interlacing methods and criteria for log-concavity, expanding understanding of combinatorial sequence behaviors.

Contribution

It provides an affirmative proof for a new type of ratio monotonicity conjecture proposed by Z. W. Sun, employing interlacing and log-concavity techniques.

Findings

Confirmed the monotonicity of the new sequence type conjectured by Sun.

Applied interlacing method for log-convexity and log-concavity analysis.

Utilized Xia's criterion for log-concavity of root sequences.

Abstract

Recently, Z. W. Sun put forward a series of conjectures on monotonicity of combinatorial sequences in the form of $\{z_n/z_{n-1}\}_{n=N}^\infty$ and $\{\sqrt[n+1]{z_{n+1}}/\sqrt[n]{z_n}\}_{n=N}^\infty$ for some positive integer $N$, where $\{z_n\}_{n=0}^\infty$ is a sequence of positive integers. Luca and St\u{a}nic\u{a}, Hou et al., Chen et al., Sun and Yang proved some of them. In this paper, we give an affirmative answer to monotonicity of another new kind of number conjectured by Z. W. Sun via interlacing method for log-convexity and log-concavity of a sequence, and we also use the criterion for log-concavity of a sequence in the form of $\{\sqrt[n]{z_n}\}_{n=1}^\infty$ due to Xia.